41
Jurnal Personalia Pelajar 17 (2014): 41 - 48
Incorrect Thinking Process Prediction for Negative Numbers Subtraction Operation Involving Positive
with Negative Integers
(Meramal Proses Kesilapan Pemikiran untuk Operasi Penolakan Nombor Negatif Melibatkan Integer
Positif dengan Negatif)
ELANGO PERIASAMY
ABSTRACT
This study was divided into two parts. The first part was to identify the incorrect answer produced by the
respondents for each item and its frequency. Then, the second part was to predict the incorrect thinking process
with respect to its frequencies that respondent might have adapted in solving such sentence questions incorrectly.
The respondents of this study were five mathematics teachers and 124 students aged 14 years old from Malaysian
secondary school. The finding shows types of mistakes made by the students for each type of items tested and
the prediction of incorrect thinking process respectively. This paper focused on the third category: subtraction
operation involving positive with negative integers. The findings revealed that in depth analysis into the students
thinking process is an essential knowledge for teachers to re-assess their teaching and correct students their
misconceptions.
Keywords: Negative numbers, subtraction, predict, teaching and learning mathematics
ABSTRAK
Kajian ini dibahagikan kepada dua bahagian. Bahagian pertama mengenal pasti kesilapan-kesilapan jawapan
yang diberikan oleh responden untuk setiap item dan kekerapannya. Kemudian, bahagian kedua meramal
kesilapan proses pemikiran responden yang memungkinkan responden mencapai penyelesaian yang salah dengan
berdasarkan kekerapannya. Responden kajian ini adalah lima orang guru Matematik dan 124 murid berusia 14
tahun daripada sekolah menengah di Malaysia. Dapatan kajian menunjukkan jenis-jenis kesilapan yang dilakukan
oleh murid untuk setiap item yang dikaji serta ramalan proses kesilapan pemikiran murid. Kertas ini berfokus
kepada kategori ketiga: operasi penolakan integer positif dengan negatif. Dapatan kajian menunjukan bahawa
analisis yang terperinci ke dalam proses pemikiran murid adalah ilmu yang penting untuk guru dalam mentafsir
semula pengajaran mereka dan memperbetulkan miskonsepsi murid.
Kata kunci: Nombor negative, penolakan, meramal, pengajaran dan pembelajaran matematik
42
INTRODUCTION
According to Lembaga Peperiksaan Malaysia (1993),
the Lower Secondary Examination (PMR) report from
the Malaysian examination Board showed that students
were unable to master the skills and understanding
the abstract concepts that involves negative number
operation in fraction, transformation and algebra.
Moreover, in the 2002 PMR examination, 47% showed
clear weaknesses in operation involving negative
number such as (-17+14), (-17+22+8), (-17-14) and
(-17+30) (Lembaga Peperiksaan Malaysia 2002).
Therefore, a study with 124 students aged 14 years old
from two secondary schools in Malaysia was carried
out by Elango Periasamy and Halimah Badioze Zaman
(2009) which revealed the existence of difficulties
in solving negative numbers subtraction operation
involving two integers. This phenomenon is explained
by Naylor (2006) as situations whereby negative
numbers extend our number line and greatly simplify
our calculations, but sometimes students struggle with
the concepts.
A review of literature showed that teachers were
very creative and innovative in teaching the concept of
subtraction and addition operation involving negative
numbers by integrating various communication tools
such as line graph, coloured stones, coloured chips,
gain-owe techniques and computer courseware in
their effort to help students acquire the knowledge of
solving negative numbers subtraction and addition
operation. These efforts shows the commitment and
creativeness of teachers that should be acknowledged
as an ongoing process that are continuously evolving
in searching ways and mean to help students acquire
knowledge related to subtraction and addition operation
in negative numbers. This was to help students avoid
arising at incorrect assumption, conclusions, thought
process and generalizations which is also important
for them to determine what things are as well as what
they are not (Brumbaugh & Rock, 2006). The first
part of this study was to identify the incorrect answer
produced by the respondents for each item and its
frequency respectively. The second part of this study
was to predict the ITP with respect to its frequencies
that respondent might have adapted in solving such
sentence questions incorrectly. The focus of this paper
was on the subtraction operation involving two single
and double positive with negative integers only.
RELATED WORKS
According to Chen and Hung (2003), a central function
of the mind is to process the information, sort them
in a meaningful way is determined by the rules and
principles employed, thus learning is then perceived as
appropriating these rules and principles and being able to
apply (or process information) according to these rules.
Therefore, the knowledge of how children construct
their early knowledge can be effectively gained from
observing and interviewing during explicit teacher set
tasks, that is if a student compute that 8 – 5 = 6, and
from examination of work samples the teacher would
immediately conclude that the child was experiencing
difficulty with the subtraction process but further
observation as the child works through examples: 7-3
= 2; 10 – 7 = 3; 2 – 1 = 4 the teacher quickly realises
the source of the errors that is the child is confusing
the digits 2 and 5 (Carnellor 2004). Moreover, a study
to refine students’ skills of addition and subtraction
including negative numbers with a seventh grade
student, turned out that errors were due to bug rules
and the lack of a critical production when executing
a purely algebraic solution were identified based on a
cognitive task analysis using several possible ways of
calculation (Terao et al., 2005).
Furthermore, findings suggest that adults’
representations of operation with negative numbers
are not as well established as their representations of
operations with positive numbers (Prather & Alibali,
2004) because in operation involving negative numbers,
some students assume many mathematical things to be
universally true and because of this they are at times,
amazed to realize their assumptions have been false
(Brumbaugh & Rock 2006). Such phenomenon was
found existed among two secondary school students
in Malaysia in solving subtraction operation involving
two integers (Elango Periasamy & Halimah Badioze
Zaman 2009). For example,
…some students are not aware that the commutative
property for addition operates in sets other than the
counting number. A series of questions or problems
like -3 + +7 = and +7 + -3 = could help lead to the
appropriate conclusions and can be amplified with
problems involving subtraction where commutativity
does not generally hold, sometimes that same students
assume to be true (-5 - +8 = and +8 - -5 =).
(Brumbaugh & Rock 2006, Pg 295)
... 3 + 3 = 6. Counting it out on your fingers can prove
the accuracy of the equation. We can see apples and
oranges in clusters of 3 or 6. It is reasonably easy to
visualize the concept of addition of positive numbers.
But, despite what all our algebra teachers have instructed
about negative numbers, when we try to add 3 apples to
43
a pile consisting of a (-3) apples, things do not work out
so simply. I get a queasy feeling in my stomach every
time I try to work with negative numbers. It makes me
quite uneasy to think that my bowl containing 3 apples
will be swept off into a vortex and lost forever if I were
to add them to a pile containing a minus 3 apples, yet the
pile of 3 apples would remain intact if I were to place
them into an empty container. The mystery of where the
3 apples would travel absolutely baffles me. And, yet, it
would be a rare mathematician who would concede that
negative numbers are an illusion. The mathematicians
don’t care if the rules and concepts they employ are
idiotic as long as they can arrive at precise answers
time after time. In other words, they know full well
that negative numbers are fraudulent, but, since they
are useful tools, they are happy to continue with the
illusion. To my way of thinking, the smallest number
of anything would have to be zero. When there are no
apples on the plate, it is empty. It would take a strange
metaphysical phenomenon indeed to allow me to place
3 apples on the plate and watch them vanish. Since
when did the sceptical people of science allow such
portals that consume apples to be considered “normal”
behaviour? This is not to say that such portals cannot
exist, but it is to say that such portals could not be called
upon to operate in a totally predictable manner each and
every time someone placed a hyphen before a number
converting it from a positive number, or something,
into a negative number, or a weird thing that is less than
nothing.
(Stanford 2003, Pg 3)
Although different strategies were used
by various researchers in helping students gain the
knowledge of solving negative numbers subtraction
operation, nevertheless real objects manipulation
for subtraction operation of negative numbers is an
illusion. Therefore, Stanford claimed that students
have been given absurd rules to apply to this weird
concept, such as a negative number when multiplied by
another negative number becomes a positive number
which is an unadulterated nonsense Stanford (2003).
Therefore, the misconceptions among students need
to be addressed via predicting their ITP. It would give
a guideline on how to hinder such misconceptions on
negative numbers subtraction operation. Moreover,
immediate practice of corrective thinking process can
be instigated and further difficulties avoid.
METHOD
The demographic information of this research was 124
respondents aged 14 years old and among them were 53
boys and 71 girls. The number of respondent achieved a
grade A is 26 (20.97%), grade B 58 (46.77%) and grade
C 40(32.26%) for their Primary School Evaluation
Examination (UPSR) in mathematics subject. The
questionnaires were divided into two sections. The first
section consists of demography data to understand the
respondent profile. The second section consists of 24
negative number subtraction operation test items with
only one correct answer for each item as in Table I. Face
validity was done with five Mathematics teachers from
three schools from a district in Malaysia. Those teachers
had an experience of teaching Negative Number topic
for at least five years. The questionnaire for this research
was created by Elango Periasamy & Halimah Badioze
Zaman (2009). A pilot test was carried out by Elango
Periasamy and Halimah Badioze Zaman (2009) with a
subject of 35 school students aged 14 years old from
a secondary school in Malaysia. The calculation of
reliability coefficient using Kuder-Richardson formula
is use for dichotomy question with right wrong answer
such as the objective questions (Alias Baba, 1999).
The Kuder-Richardson (KR20) reliability estimation
value of this instrument was 0.919544. The reliability
was calculated using the KR20 formula (Mervis &
Spagnolo, 1995) with Microsoft Office Excel 2007.
According to Mervis and Spagnolo (1995), when the
test format has only one correct answer then KR20 is
algebraically equivalent to Cronbach alpha. Therefore,
in this case the KR20 reliability estimation value of this
pilot test is equivalent to Cronbach alpha coefficient.
The 24 items of this study (as in Table 1) were
rearranged into four categories by Elango Periasamy
and Halimah Badioze Zaman (2011) as follows:
i. First category – Subtraction Operation Involving
Two Positive Integers
ii. Second category – Subtraction Operation Involving
Negative with Positive Integers
iii. Third category – Subtraction Operation Involving
Positive with Negative Integers
iv. Forth category –Subtraction Operation Involving
Negative with Negative Integers
Therefore, each category consists of 6 items
with respect to its theme. The focus of this paper was
limited to the third category to share and predict the
ITP of subtraction operation involving a positive with a
negative integers (a – b, a>0, b<0) only.
Therefore, each category consists of 6 items with
respect to its theme. The focus of this paper was
limited to the third category to share and predict the
44
TABLE 1. Negative number subtraction operation test items
No
Item
No
Item
1
5-2=
13
-8 - 13 =
2
3
-5 - 2 =
-5 - (-2) =
14
15
8 - 13 =
-8 - (-13) =
4
5 - (-2) =
16
8 - (-13) =
5
-2 - 5 =
17
16 - 23 =
6
7
2-5=
-2 - (-5) =
18
19
-16 - 23 =
-16 - (-23) =
8
2 - (-5) =
20
16 - (-23) =
9
13 - 8 =
21
-23 - 16 =
10
-13 - 8 =
22
23 - 16 =
11
-13 - (-8) =
23
-23 - (-16) =
12
13 - (-8) =
24
23 - (-16) =
ITP of subtraction operation involving a positive with a
negative integers (a – b, a>0, b<0) only.
In conjunction, this study was divided into two
parts. The first part was to identify the incorrect answer
produced by the respondent for each item and its
frequency for the third category. Then, the second part
was to predict the ITP with respect to its frequencies
that that respondent might have adapted in solving such
sentence questions incorrectly. Therefore, all possible
ITP that student would have used in order to arrive at
those incorrect answers need to be derived explicitly
by analyzing students prior knowledge and teachers
teaching approach for negative numbers subtraction
operation that might have been responsible for such
conflict in adapting the correct thinking or rules in
solving subtraction operation. Then, re-confirming with
five mathematics teachers.
FINDINGS
The first part was to identify the incorrect answer
produced by the respondent for each item and its
frequency. Table 2 shows the result of the first part of
this study. The highest incorrect solution was for item 16
(46.77%) whereby 2, 30 and 26 students gave incorrect
solution 5, -5 and -21 respectively; followed by item 12
(43.55%) whereby 33, 15 and 6 students gave incorrect
solution 5, -21 and -5 respectively; then item 4 (41.13%)
whereby 25, 6 and 20 students gave incorrect solution
3, -3 and -7 respectively; for item 20 meanwhile 15, 8,
24 and 2 students gave incorrect solution -39, 7, -7 and
-8 respectively; continued by item 8 (36.29%) then 10,
17 and 18 students gave incorrect solution 3, -3 and 7
respectively; and finally for item 24 (33.8%) whereby
12, 18 and 12 students gave incorrect solution -39, -7
and 7 respectively.
45
TABLE 2. Subtraction of positive with negative integer
No
Item
4
5 – (-2) =
8
2 – (-5) =
12
13 – (-8) =
16
8 – (-13) =
20
16 – (-23) =
24
•
23 – (-16) =
Incorrect Solution
Frequency
Total (%)
51 (41.13%)
3
25
-3
6
-7
20
3
-3
-7
5
10
17
18
33
-21
15
-5
6
5
2
-5
30
-21
26
-39
15
7
8
-7
24
-8
2
-39
12
-7
18
7
12
The S3T1 type
The S3T1 type addressed the subtraction operation
where the magnitude value of the first positive integer
was greater than the magnitude value of the second
45 (36.29%)
54 (43.55%)
58 (46.77%)
49 (39.52%)
42 (33.8%)
negative integer. An example of the S3T1 type item is
15- (-7). Based on this item, a prediction of respondents’
ITP for S3T2 type item was made and discussed in
Table 3.
TABLE 3. Predict S3T1 Type ITP
ITP
Types of Solution
Predict ITP
1
15- (-7) = -8
2
15- (-7) = -22
3
15- (-7) = 8
Move 1: positive sign in front of number 15 multiply negative sign in front of
number 7 which gives the sign for final answer, in this case negative. Move 2:
Now the question is rewritten as 15-7= which gives 8 as its solution. Then, with
reference to move 1, the answer becomes -8, thus 15- (-7) = -8
Move 1: Positive sign in front of number 15 multiply negative sign in front of
number 7 which gives the sign for final answer, in this case negative. Move
2: There exist two negative signs in between 15 and 7, thus negative multiply
negative is positive. Then perform 15 + 7 which gives 22. Now, from move 1 the
final answer will have negative sign. Thus, 15-(-7) = -22.
Move 1: Positive sign in front of number 15 multiply negative sign in front of
number 7 which gives a negative sign. Then, multiply it with the negative sign
in between 15 and -7 which gives the sign for final answer, in this case positive.
Now the question is rewritten as 15 – 7. Move 2: Perform 15 – 7 which gives 8
as its solution. Now from move 1 the final answer will have negative sign. Thus,
15- (-7) = 8
46
•
The S3T2 type
The S3T2 type addressed the subtraction operation
feature where the magnitude value of the first positive
integer was smaller than the magnitude value of the
second negative integer. An example of the S3T2 type
item is 7- (-15). Based on this item, a prediction of
respondents’ ITP for S3T2 type item was made and
discussed in Table 4.
TABLE 4. Predict S3T2 type ITP
ITP
Types of Solution
Predict ITP
1
7- (-15) = 8
2
7- (-15) = -22
3
7- (-15) = -8
Move 1: Positive sign in front of number 7 multiply negative sign in front of
number 15 which give negative sign. Then, multiply it with the negative sign in
between 7 and -15. Now the question is rewritten as 7 – 15. Now the question
becomes a first category type of sentence question. Move 2: Number 15 is bigger
than 7. Then perform 15 – 7 which give 8, thus 7 – (-15) = 8.
Move 1: Positive sign in front of number 7 multiply negative sign in front of
number 15 which gives the sign for final answer, in this case negative. Move
2: There exist two negative sign in between 7 and 15, thus negative multiply
negative is positive. Then, rewrite sentence question as 7 + 15 and perform 7 +15
which gives 22. Now, from move 1 the final answer will have negative sign, thus
7- (-15) = -22
Move 1: positive sign in front of number 7 multiply negative sign in front of
number 15 which gives the sign for final answer, in this case negative. Move 2:
Now the question is rewritten as 7-15 (first category type sentence question) but
performs 15 -7 which gives 8 as its solution. Then, with reference to move 1, the
answer becomes -8, thus 7- (-15) = -8
DISCUSSION
Items 4, 8, 12, 16, 20 and 24 from the S3 group which
were the subtraction operation feature of a positive
number with a negative number in parenthetical. Items
4, 12 and 24 were the subtraction operation feature of
a positive integer number which had greater value with
a negative integer number which had smaller absolute
value in parenthetical (a – b =, a>0, b<0, a>|b|). The
research findings showed that the respondents’ use of
the three different thinking process techniques gave
rise to wrong answers. Among the three thinking
process techniques, the ITP 3 technique was found to
be more dominant followed by the ITP 2 and the ITP
1, respectively. While, items 8, 16 and 20 from the
S3T2 group were the subtraction operation feature of a
positive integer number which had a smaller value with
a negative integer number that had a larger absolute
value in parenthetical (a – b =, a>0, b<0, a<|b|). The
research findings showed that the respondents’ use of
the three different thinking process techniques gave
rise to wrong answers. Among the three thinking
process techniques, the ITP 3 technique was found to
be more dominant followed by the ITP 2 and the ITP
1, respectively. Therefore, these incorrect thinking
processes created opportunities for teachers to reflect
on their teaching and learning process on the specific
domain.
The process of predicting ITP was a very
tedious and time consuming. The process needed
special diagnostic sentence questions which could
create conflict in the students’ thinking process in
solving them and with proper analysis and synthesis
when predicting ITP and followed by reconfirming the
prediction. Nevertheless, this study was an interesting
experience towards exploring the ITP respondents
acquired, moreover the findings can be important in
helping mathematics educators to be aware of such ITP
could exist and proper precaution should be taken into
consideration during teaching and learning of negative
numbers subtraction operation or remedial works. It
is because such misconceptions firstly, interfere with
learning when students use them to interpret new
experiences and secondly, students are emotionally and
intellectually attached to their misconceptions because
they have actively constructed them and students give
up their misconceptions, which can have such a harmful
effect on learning, only with great reluctance (Rose et
al., 2007). But to teach in a way that avoids creating and
misconceptions is not possible and we have to accept
that students will make some incorrect generalizations
that will remain hidden unless the teacher makes
47
specific efforts to uncover them (Askew & William,
1995) in Rose et al., (2007).
Therefore, according to Carnellor (2004), there
is no simple one answer to guide specific practice and
teachers must provide a wide variety of methods through
their diverse repertoire of class room practices in their
lesson planning, the topic presented, the instructional
experiences and activities incorporated in the learning
session and their responses to children’s questions.
Moreover, Brumbaugh and Rock (2006) suggested that
assistance is provided to the discovery process through
a carefully developed set of problems that guide the
student to appropriate responses. Nevertheless, Chen
and Hung (2003) said that by analysing the way the
experts think and by teaching students these expert
ways of thinking, cognitivists hope to instruct students
in order to emulate expert thinking and develop the
students’ expertise is a particular domain of knowledge.
CONCLUSION
In conjunctions, this study was to identify and predict
the ITP of respondents in solving negative numbers
subtraction operation involving two integer sentence
questions limited to single double digit integers.
Even though, all students in a class room are taught
equally and simultaneously but the way they perceive
and process the knowledge are in their own unique
way should be acknowledged with great enthusiasm.
Therefore, the most sadness of this study was that the
teachers and students were unaware of the existence of
the ITP until a study of this kind was conducted. AskellWilliams et al. (2007) perceives it as a situation where
students develop robust mental models of teaching
and learning during their school years, and as such,
often teach as they were taught-possibly perpetuating
practices that limit intellectual inquiry in classrooms.
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48
Elango Periasamy (Dr.)
IPG Kampus Pendidikan Teknik
Kompleks Pendidikan Nilai,
71760 Bandar Enstek,
Nilai, Negeri Sembilan.
E-mel: surensutha@yahoo.com